We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner T(k−1, k, v) bitrades, extended 1perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-...

While a spanning tree spans all vertices of a given graph, a Steiner tree spans a given subset of vertices. In the Steiner minimal tree problem, the vertices are divided into two parts: terminals and nonterminal vertices. The terminals are the given vertices which must be included in the solution. The cost of a Steiner tree is defined as the total edge weight. A Steiner tree may contain some no...

Given a complete graph G = (V,E), where each vertex is labeled either terminal or Steiner, a distance function d : E → R, and a positive integer k, we study the problem of finding a Steiner tree T spanning all terminals and at most k Steiner vertices, such that the length of the longest edge is minimized. We first show that this problem is NP-hard and cannot be approximated within a factor 2− ε...

In this paper, the Wiener polynomials Wn(G;x) for some special graphs G including path graphs and cycle graphs are obtained. Moreover, for vertex-disjoint connected graphs G1 and G2, formulas for the Wiener polynomials of Steiner n-distance of compound graphs G1•G2 and G1:G2 are obtained in terms of those polynomials for G1and G2.

This paper presents a fast polynomial approximation algorithm for constructing the Steiner tree of minimum total cost (the cost of the edge (i; j) equal to the rectilinear distance between these nodes) with diierent requirements on delays along each path to destination. The exibility of the proposed algorithm permits to drive the number of Steiner nodes in the solution without increasing the tr...

A doubly constant weight code is a binary code of length n1 + n2, with constant weight w1 + w2, such that the weight of a codeword in the first n1 coordinates is w1. Such codes have applications in obtaining bounds on the sizes of constant weight codes with givenminimum distance. Lower and upper bounds on the sizes of such codes are derived. In particular, we show tight connections between opti...

Given a connected, undirected distance graph with required classes of nodes and optional Steiner nodes, nd a shortest tree containing at least one node of each required class. This problem called Class Steiner Tree is NP-hard and therefore we are dependent on approximation. In this paper, we investigate various restrictions of the problem comparing their complexities with respect to approximabi...

We describe a technique to improve Smith’s branch-and-bound algorithm for the Euclidean Steiner tree problem inRd . The algorithm relies on the enumeration and optimization of full Steiner topologies for corresponding subsets of regular points. We handle the case of two Steiner points colliding during the optimization process—that is when there is a small enough distance between them—to dynamic...

Given a signal net N = s; 1; ;n to be the set of nodes, with s the source and the remaining nodes sinks, an MRDPT (minimum-cost rectilinear Steiner distance-preserving tree) has the property that the length of every source to sink path is equal to the rectilinear distance between the source and sink. The minimum-cost rectilinear Steiner distance-preserving tree minimizes the total wire length w...